Quantum Uncertainty Relations for Thermodynamic Energy Flows

Abstract

The Heisenberg uncertainty relation, which links the uncertainties of the position and momentum of a particle, has an important footprint on the quantum behavior of a physical system. Analogous to this principle, we propose that thermodynamic currents associated with work, heat, and internal energy satisfy their own uncertainty relations. To formalize this idea, we represent these currents by well-defined Hermitian operators, constructed so that their expectation values match the corresponding average currents. Because these operators generally do not commute, the resulting quantum currents differ fundamentally from their classical counterparts. Using the Robertson-Schrödinger uncertainty relation, we derive various uncertainty relations that link different thermodynamic flows. We further illustrate this approach by applying it to quantum batteries, where we derive an energy-power uncertainty relationship and show how measurements affect the fluctuations.

Figures (7)

• Figure 1: A pictorial representation of the non-commutativity of the energy ($\bm{E}_B$) and power ($\bm{P}_B$) operators for quantum batteries, and the power (${\pmb{\mathring{\mathcal{W}}}}$) and heat flow (${\pmb{\mathring{\mathcal{Q}}}}$) operators in an open system. The precision of measurements of $\bm{E}_B$ and $\bm{P}_B$ for the battery (or ${\pmb{\mathring{\mathcal{W}}}}$ or ${\pmb{\mathring{\mathcal{Q}}}}$ for more general open quantum dynamics) are lower bounded by the degree of non-commutativity of these operators.
• Figure 2: Pictorial representation of a system of two interacting one-dimensional harmonic oscillators [see Eq. \ref{['eq:two_interacting_harmonic_oscillators']}] . The first oscillator (left) is regarded as the system, with the second one as the environment. Work is done to the system by changing the frequency $\omega_a$ as a function of time and heat is exchanged by the system with the second oscillator due to the position-position coupling.
• Figure 3: Numerically computed operator variances ($\sigma_{\pmb{\mathring{\mathcal{W}}}}, \sigma_{\pmb{\mathring{\mathcal{Q}}}}$ and $\sigma_U$) and the corresponding lower bounds obtained using the Robertson-Schrödinger uncertainty relation for a system of two interacting spins with a Hamiltonian \ref{['eq:two_interacting_spins_example']}. The left panel corresponds to $f(t)=2\exp(-t/2)$, while the right panel corresponds to $f(t)=\sin(t) + 2$, with $\hbar=g=1$ in both cases. The upper panel shows numerically computed $\sigma_{\pmb{\mathring{\mathcal{W}}}}\sigma_{\pmb{\mathring{\mathcal{Q}}}}$ as well as the lower bound from Eq. \ref{['eq:lower_bound_sigmas_of_qdot_wdot_two_spins']}. The lower panels show $\sigma_U$ as well as lower bounds on it obtained using Eq. \ref{['eq:lower_bound_on_sigma_u_version_qdot']} and Eq. \ref{['eq:lower_bound_on_sigma_u_version_wdot']}. In all cases, the initial state was chosen to be $\ket{\psi(0)} = \ket{\uparrow} \otimes \ket{\uparrow}$, with $\ket \uparrow$ representing the up state along the $z$ direction. We note that the curve for $\sigma_{\bm{U}}$ coincides with the bound obtained using Eq. \ref{['eq:lower_bound_on_sigma_u_version_wdot']}.
• Figure 4: Numerically computed operator variances ($\sigma_{\pmb{\mathring{\mathcal{W}}}}, \sigma_{\pmb{\mathring{\mathcal{Q}}}}$ and $\sigma_U$) and the corresponding lower bounds obtained using the Robertson-Schrödinger uncertainty relation for a system of two driven, interacting oscillators with a Hamiltonian \ref{['eq:two_interacting_harmonic_oscillators']}. The left panel corresponds to $\omega_a(t)=2\exp(-t/2)$, while the right panel corresponds to $\omega_a(t)=\sin(t) + 2$, with $\hbar=g=m=\omega_b=1$ in both cases. The upper panel shows numerically computed $\sigma_{\pmb{\mathring{\mathcal{W}}}}\sigma_{\pmb{\mathring{\mathcal{Q}}}}$ as well as the lower bound from Eq. \ref{['eq:lower_bound_sigmas_of_qdot_wdot_two_oscillators']}. The lower panels show $\sigma_U$ as well as lower bounds on it obtained using Eq. \ref{['eq:lower_bound_on_sigma_u_version_qdot']} and Eq. \ref{['eq:lower_bound_on_sigma_u_version_wdot']}. In all cases, the initial state was chosen to be $\ket{\psi(0)} = \ket{0} \otimes \ket{0}$, i.e. the tensor product of the ground states of the two oscillators.
• Figure 5: Energy-power uncertainty for a two-level system described by \ref{['eq:isolated_single_qubit_H']} without (left figure) and with two equally-spaced measurements in the $\sigma^3$ basis (right figure). Upper panel: Evolution of average energy and average power are shown in solid lines (blue and red respectively), and the width of the shaded region represents the uncertainty $2 \sigma_{E_B}$ and $2\sigma_{P_B}$. Specifically, we plot $\langle E_B \rangle$ sandwiched between $\langle E_B \rangle \pm \sigma_{E_B}$ and similarly for $P_B$. Lower panel: The product $\sigma_{P_B}^2 \sigma_{E_B}^2$ is shown along with the lower bound \ref{['eq:energy_power_uncertainty_closed_battery']}. We show the commutator term separately, along with the bound that includes the covariance term. The simulations correspond to the values $h_0=1.2, h_3 = 0.2$, $v_0=0$ and $\vec{v} = (0.5,0.6,0)$. The initial condition $\rho(t=0)$ was chosen by setting $\vec{\beta} = (0,0,0.5)$ in \ref{['eq:rho_0_for_isolated_spin']}, i.e. the initial condition corresponds to the spin-down state in the $\bm{\sigma}^z$ basis. We set $\hbar=1$ for convenience.